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Full text of "Treatise On Analysis Vol-Ii"

334      XV   NORMED ALGEBRAS AND SPECTRAL THEORY

4. BANACH ALGEBRAS WITH INVOLUTION. STAR ALGEBRAS

Let A be an algebra over the field C of complex numbers. An involution on
A is a bijection x*-+x* of A onto A, satisfying the following conditions:

(15.4.1)     (x*)* = x,     (x + y)* = x* + /",      (Ax)* = Ix*,      (xj;)* = y*x*

for all x, y in A and all A e C. The element x* is often called the adjoint of x,
and a subset of A which is stable under the involution is said to be selfi

adjoint.

If A has a unit element e, then x - e* = (e - x*)* = x** = x, and similarly
e* . jc = x for all x 6 A, by virtue of (15.4.1). Hence

(15.4.2)                                              e* = e.

If x is invertible in A, then (x"""1)*** = (xx""1)* = e* = e, and similarly
x^x'1)* = e> by (15.4.1) and (15.4.2). Hence x* is invertible, and

(15.4.3)                                 (x-1)* = (x*)"1.

A normed algebra with involution is a normed algebra A equipped with an
involution xi>x* such that

(15.4.4)                                        ||x*|| = ||x||

for all x e A.

A star algebra is a Banach algebra A equipped with an involution xh-x*
such that

(15.4.5)                                        ||x||2 = ||x*x||
for all x 6 A.

A star algebra satisfies (15.4.4), for it follows from (15.4.5) and (15.1.1)
that ||x||2 g ||x||  ||x*||, whence ||x|| ^ ||x*|| for all xe A; interchanging x and
x*, we obtain (15.4.4).

Examples of Banach Algebras with Involution and Star Algebras

(15.4.6) If X is a metrizable compact space, the Banach algebra ^C(X) is a
star algebra with respect to the involution/W/ (the complex conjugate of/).
If X consists of a single point, #C(X) is isomorphic to the field C.itely differentiable complex-valued functions on [0, 1].